What are the generators of $\mathbb C[V^m]^W$, where $W$ is the Weyl group of type $E_6, E_7, E_8$, V^m denote 'm' (m > 1) copies of the Cartan subalgebra and the action is the diagonal action?

Is there any reference where I can find the generators explicitly?

  • 1
    $\begingroup$ Are there any restrictions on generators that you want? Why exceptional types? This is a very hard problem even for $A_n$ (symmetric group), at least, if you want a minimal system of generators. $\endgroup$ – Victor Protsak May 13 '10 at 20:53
  • $\begingroup$ Looks like another shooter in the dark... $\endgroup$ – Victor Protsak May 14 '10 at 4:08
  • $\begingroup$ When m=1, if the Weyl group is from a classical Lie algebra then this is a known result, and it is not hard to imagine that somewhere the case of m copies has been worked out. I suspect this is why the asker is more interested in exceptional type. $\endgroup$ – Q.Q.J. May 15 '10 at 4:44
  • $\begingroup$ No, any system of generator will work for me, not necessarily a minimal system. For type $A_n, B_n, C_n, D_n$ and $G_2$ I know a set of generators but I do not have any clue for other exceptional types. Actually I am much more interested in the degrees of the generators. $\endgroup$ – user6079 May 15 '10 at 11:48
  • $\begingroup$ @Q.Q.J. Could you please give me a reference for you claim concerning the case $m=1$? $\endgroup$ – Lepanais Feb 3 '14 at 19:14

I don't think the answer is known. The paper [Hunziker, Classical invariant theory for finite reflection groups. Transform. Groups 2 (1997), no. 2, 147–163] is relevant. The author conjectures an answer and shows his answer is correct for $F_4$.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.